Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135524 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
976 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 976 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,14,7)(4,15,17,8)(5,9)(6,10)(12,27,29,22)(13,25,31,21)(16,24,18,32)(19,23)(20,28,26,30);(1,3)(2,7)(4,13)(5,14)(6,12)(8,21)(9,11)(10,22)(15,20)(16,31)(17,28)(18,30)(19,29)(23,27)(24,26)(25,32);(2,8,23,32)(3,12)(4,16)(7,20,27,26)(9,24,10,15)(11,25,22,21)(13,30)(14,29)(17,18)(28,31)Polycyclic presentation:
2^9|a^2=f,c^2=dh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,e^c=egi,f^c=fh,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72237414, #72237443, #72237445, … |
| maximal normal subgroup of | #72237414, #72237443, #72237445, … |
| maximal quotient of | #72338952 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42720);